All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance

TL;DR

This work establishes a general sufficient condition under which chemical distance $\operatorname{d}$ in spatial random graphs is linearly bounded below by the Euclidean distance $|x-y|$, by controlling long-range effects through two polynomial criteria: scarcity of long edges and weak large-distance correlations. Using a Berger-type renormalisation scheme, the authors show that if the graph satisfies polynomial mixing with exponent $\xi<0$ and a no-long-edges condition with exponent $\mu<-d$, then there exists $\eta>0$ such that $\mathbb{P}(\neg \mathcal{D}_{L}^{\eta}(m))$ decays at rate $\le \xi \vee (d+\mu)$, and in the regime $d+\mu>\xi$ this decay is governed by the long-edge probability $\mathbb{P}(\mathcal{L}(m,m))$. Consequently, either any crossing must use a path of length proportional to Euclidean distance or a single spanning edge occurs, and with an appropriate scale choice one obtains almost-sure linear lower bounds on chemical distance, recovering Berger (2004) and extending it to a broad class of translation-invariant, locally finite graphs. The framework is demonstrated on several models, including the weight-dependent random connection model, the soft Boolean model with local interference, and ellipses percolation, and is shown to apply to more general vertex-location schemes such as Cox and Gibbs processes. This yields quantitative insights into long-range percolation by linking the decay of short-path probabilities to the presence (or absence) of long edges, and highlights conditions under which linear lower bounds on chemical distance hold in diverse spatial networks.

Abstract

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

Figures (2)

• Figure 1: The bad box $B_n$, its sub boxes, and, dotted, the right-shifted box $B_n^{(1, 0)}$ with its sub boxes. In grey, the bad sub boxes of $B_n^{(1, 0)}$. On the left, two long edges are shown. The longer edge is too long so that $B_n$ fails Property (ii) (a). The second longest edge is short enough on scale-$n$ but, on scale-$(n-1)$, it creates a bad sub box of the right-shifted box. On the right, $B_n$ fails Property (ii) (b) as the right-shifted box contains $3^d+1$ bad sub boxes.
• Figure 2: A path connecting $x$ and $y$ inside a large good box, with the bad sub boxes in grey. The path decomposes into a good segment $\pi_1$ from $x$ to the last vertex before entering $Q_1$; the bad segment $\sigma_1$, which is not shown in the picture, consisting of the path segment between the first entrance to $Q_1$ and the last exit of $Q_1$; as the path directly enters $Q_3$ after leaving $Q_1$, the good segment $\pi_2$ is empty; the bad segment $\sigma_2$, again not shown, consists of the path segment between the first entrance to $Q_3$ and the last exit of $Q_3$; the final good segment $\pi_3$ then connects the vertex after leaving $Q_3$ to $y$. As the depicted part of the path is completely contained in good regions, no long edges can be used.

Theorems & Definitions (16)

• Remark 1
• Theorem 1: Linear graph distances, lower bound
• Remark 2: On the upper bound
• Lemma 2
• proof
• Corollary 3
• Definition 4: Good boxes
• Lemma 5
• Corollary 6
• Proposition 7